Prime forms involving Repunits, R_n.

Thu Oct 14 22:58:21 CEST 2004

Et al,

	There is a non conformity about the naming and presentation of these
sequences. I would like this situation rectified and the sequences extended.
Also they ought to be cross referenced or better, referenced to the index.
I am writing you for suggestions and help in extending these sequences. At
the minimum, we should be able to get to exponents 10000. There are 57
sequences below, A004023 is presented twice, of which only A004023, A096507
& A089675 have been searched to that limit. Personally I can get to >6500 by
letting my computer run over night and that is on a 1.2GHz Athlon chip.

	I would be willing to act as the collection point on this if that is OK
with every one. If we go about this project wisely we should have this done
inside two weeks.

	Concerning the %Y cross reference lines, hyphenated sequences only allow
you the hyper link to the two extremes, so I generally do not like hyphens.

Sequentially yours,

Bob.

	Prime forms involving Repunits, R_n.

The 29 forms, X*Repunits+/-Y, (X,Y)=1, X+/-Y <10,
			X&Y belong to {d} d being the digits 1..9.

Form 1Rn+0
%I A004023 M2114
%S A004023 2,19,23,317,1031,49081,86453
%N A004023 Prime "repunits": 11...111 = (10^n - 1)/9 is prime.
............
%H A004023 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/11111.htm">Factorizations of 
11...11 (Repunit)</a>.
............
%Y A004023 See A004022 for the actual primes.
%K A004023 hard,nonn,nice
%O A004023 1,1
%A A004023 njas

Form 1Rn+2
%I A097683
%S A097683 0,1,2,3,5,9,11,24,84,221
%N A097683 Numbers n such that (10^n-1)/9 + 2 is prime.
%C A097683 Values indicate primes of the form "(n-1) ones followed by a three"; 
zero is a degenerate case. Related to the base 10 repunit primes.
%t A097683 Do[ If[ PrimeQ[(10^n - 1)/9 + 2], Print[n]], {n, 0, 1000}] (from RGWv 
Oct 14 2004)
%Y A097683 Cf. A004023, A097684, A097685.
%K A097683 hard,more,nonn
%O A097683 0,3
%A A097683 Carl R. White (cyrek(AT)cyreksoft.yorks.com), Aug 19 2004

Form 1Rn+6
%I A097684
%S A097684 1,2,4,5,8,23,29,40,131,136,215,611,767,2153,2576
%N A097684 Numbers n such that (10^n-1)/9 + 6 is prime.
%C A097684 Values indicate primes of the form "(n-1) ones followed by a seven". 
Related to the base 10 repunit primes.
%C A097684 2153 and 2576 produce probable primes. - a(12)-a(15) from Rick L. 
Shepherd (rshepherd2(AT)hotmail.com), Aug 23 2004
%F A097684 a(n) = A056655(n) + 1 for all n >= 0. - a(12)-a(15) from Rick L. 
Shepherd (rshepherd2(AT)hotmail.com), Aug 23 2004
%o A097684 (PARI) for (i=1,1000,if(isprime((10^i-1)/9 + 
6),print1(i,","),print1("."))) (Bouayoun)
%t A097684 Do[ If[ PrimeQ[(10^n - 1)/9 + 6], Print[n]], {n, 0, 2600}] (from RGWv 
Oct 14 2004)
%Y A097684 Cf. A004023, A056655, A097683, A097685.
%K A097684 nonn
%O A097684 0,2
%A A097684 Carl R. White (cyrek(AT)cyreksoft.yorks.com), Aug 19 2004
%E A097684 a(12) and a(13) from Mohammed Bouayoun 
(mohammed.bouayoun(AT)sanef.com), Aug 23 2004.
%E A097684 a(12)-a(15) from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Aug 23 2004

Form 1Rn+8
%I A097685
%S A097685 2,5,6,8,17,50,684,720,1452,1679
%N A097685 Numbers n such that (10^n-1)/9 + 8 is prime. Equals A056659 + 1. See 
A056659, the main entry for this problem, for additional information.
%C A097685 Values indicate primes of the form "(n-1) ones followed by a nine". 
Related to the base 10 repunit primes.
%t A097685 Do[ If[ PrimeQ[(10^n - 1)/9 + 6], Print[n]], {n, 0, 1700}] (from RGWv 
Oct 14 2004)
%Y A097685 Equals A056659 + 1. Cf. A004023, A097683, A097684.
%K A097685 nonn
%O A097685 1,1
%A A097685 Carl R. White (cyrek(AT)cyreksoft.yorks.com), Aug 19 2004

Form 2Rn-1
%I A084832
%S A084832 4,18,100,121,244,546,631,1494
%N A084832 Numbers n such that 2/9*(-1+10^n)-1 is prime.
%C A084832 Larger values not certified prime.
%e A084832 a(1)=4 because 2/9*(-1+10^4)-1 = 2221 is prime.
%e A084832 n=18 means that 222222222222222221 is prime.
%t A084832 Do[ If[ PrimeQ[(10^n - 1)/9 + 6], Print[n]], {n, 0, 1700}] (from RGWv 
Oct 14 2004)
%Y A084832 Cf. A084831, A096503, A096504, A096505, A096506, A096507, A096508, 
A096841, A096842, A096843, A096844, A096845, A096846, A000203.
%K A084832 more,nonn
%O A084832 1,1
%A A084832 Jason Earls (jcearls(AT)cableone.net), Jun 05 2003
%E A084832 a(8) from Labos E. (labos(AT)ana1.sote.hu), Jul 15 2004

%I A096844 is the same as A084832, so simply delete.

Form 2Rn+1
%I A096506
%S A096506 1,2,3,8,11,36,95,101,128,260,351,467,645
%N A096506 Exponents n for which 1+2*(10^n-1)/9 is a prime.
%e A096506 n=36: 222222222222222222222222222222222223 is a prime number.
%t A096506 Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 1], Print[n]], {n, 1000}] (from RGWv 
Oct 14 2004)
%Y A096506 Cf. A096503, A096504, A096505, A096506, A096507, A096508.
%K A096506 base,nonn
%O A096506 1,2
%A A096506 Labos E. (labos(AT)ana1.sote.hu), Jul 12 2004

Form 2Rn+5
%I A099409
%S A099409 1, 3, 9, 15, 28, 64
%N A099409 Numbers n such that 2(10^n-1)/9 + 5 is prime.
%t A099409 Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 5], Print[n]], {n, 1000}]
%Y A099409 Cf. .
%O A099409 0,2
%K A099409 nonn
%A A099409 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 2Rn+7
%I A099410
%S A099410 2, 3, 5, 14, 176, 416
%N A099410 Numbers n such that 2(10^n-1)/9 + 7 is prime.
%t A099410 Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 7], Print[n]], {n, 1000}]
%Y A099410 Cf. .
%O A099410 0,2
%K A099410 nonn
%A A099410 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 3Rn-2
%I A055557
%S A055557 2,3,4,5,6,7,8,18,40,50,60,78,101,151,319,382,784,1732,1918
%N A055557 Numbers n such that (10^n-7)/3 is prime.
%C A055557 n also gives the number of decimal digits.
%C A055557 Sierpinski attributes the primes for n = 2,...,8 to A. Makowski.
%D A055557 W. Sierpinski, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from 
the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.
%H A055557 Dave Rusin, <a 
href="http://www.math.niu.edu/~rusin/known-math/98/exp_primes">Primes in 
exponential sequences</a>
%t A055557 Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 1, 1918}]
%o A055557 (PARI) for(n=1,2000, if(isprime((10^n-7)/3),print(n)))
%Y A055557 Cf. A051200, A033175.
%K A055557 nonn
%O A055557 2,1
%A A055557 Labos E. (labos(AT)ana1.sote.hu), Jul 10 2000
%E A055557 Corrected and extended by Jason Earls (jcearls(AT)cableone.net), Sep 22 
2001

Form 3Rn+4
%I A099411
%S A099411 1, 2, 3, 6, 46, 394, 978
%N A099411 Numbers n such that 3(10^n-1)/9 + 4 is prime.
%t A099411 Do[ If[ PrimeQ[ 3(10^n - 1)/9 + 4], Print[n]], {n, 1000}]
%Y A099411 Cf. .
%O A099411 1,2
%K A099411 nonn
%A A099411 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 4Rn-3
%I A099412
%S A099412 0, 2, 4, 11, 28, 55, 94, 475
%N A099412 Numbers n such that 4(10^n-1)/9 - 3 is prime.
%t A099412 Do[ If[ PrimeQ[ 4(10^n - 1)/9 - 3], Print[n]], {n, 0, 1000}]
%Y A099412 Cf. .
%O A099412 0,2
%K A099412 nonn
%A A099412 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 4Rn-1
%I A096845
%S A096845 1,2,3,6,9,12,30,32,183,297,492
%N A096845 Exponents n for which -1+4*(10^n-1)/9 is a prime.
%e A096845 n=30 means that 444444444444444444444444444443 is prime.
%t A096845 Do[ If[ PrimeQ[ 4(10^n - 1)/9 - 1], Print[n]], {n, 1000}] (from RGWv 
Oct 14 2004)
%Y A096845 Cf. A096503-A096508, A096841-A096846, A000203.
%K A096845 more,nonn
%O A096845 1,2
%A A096845 Labos E. (labos(AT)ana1.sote.hu), Jul 15 2004

Form 4Rn+3
%I A099413
%S A099413 0, 1, 2, 4, 10, 20, 26, 722
%N A099413 Numbers n such that 4(10^n-1)/9 + 3 is prime.
%t A099413 Do[ If[ PrimeQ[ 4(10^n - 1)/9 + 3], Print[n]], {n, 1000}]
%Y A099413 Cf. .
%O A099413 0,3
%K A099413 nonn
%A A099413 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 4Rn+5
%I A099414
%S A099414 0, 3, 5, 6, 48, 108, 245
%N A099414 Numbers n such that 4(10^n-1)/9 + 5 is prime.
%t A099414 Do[ If[ PrimeQ[ 4(10^n - 1)/9 + 5], Print[n]], {n, 1000}]
%Y A099414 Cf. .
%O A099414 0,3
%K A099414 nonn
%A A099414 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 5Rn-4
%I A099415
%S A099415 12, 13, 609
%N A099415 Numbers n such that 5(10^n-1)/9 - 4 is prime.
%t A099415 Do[ If[ PrimeQ[ 5(10^n - 1)/9 - 4], Print[n]], {n, 1000}]
%Y A099415 Cf. .
%O A099415 0,1
%K A099415 nonn
%A A099415 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 5Rn-2
%I A099416
%S A099416 0, 1, 2, 8, 26, 66, 74, 233, 473, 540
%N A099416 Numbers n such that 5(10^n-1)/9 - 2 is prime.
%t A099416 Do[ If[ PrimeQ[ 5(10^n - 1)/9 - 2], Print[n]], {n, 0, 1000}]
%Y A099416 Cf. .
%O A099416 0,3
%K A099416 nonn
%A A099416 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 5Rn+2
%I A099417
%S A099417 0, 1, 3, 4, 6, 10, 15, 22, 88, 207, 528, 960
%N A099417 Numbers n such that 5(10^n-1)/9 + 2 is prime.
%t A099417 Do[ If[ PrimeQ[ 5(10^n - 1)/9 + 2], Print[n]], {n, 0, 1000}]
%Y A099417 Cf. .
%O A099417 0,3
%K A099417 nonn
%A A099417 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 5Rn+4
%I A099418
%S A099418 2, 8, 12, 18, 26, 32, 138, 188, 222, 338
%N A099418 Numbers n such that 5(10^n-1)/9 + 4 is prime.
%t A099418 Do[ If[ PrimeQ[ 5(10^n - 1)/9 + 4], Print[n]], {n, 0, 1000}]
%Y A099418 Cf. .
%O A099418 0,3
%K A099418 nonn
%A A099418 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 6Rn-5
%I A098088
%S A098088 2,3,4,10,18,21,22,28,43,66,121,133,178,241,454,553,1600,2175,2978,3649
%N A098088 Integers n such that ((2*10^n)-17)/3 is a prime number.
%C A098088 No others less than 7000. n = 1600, 2175, 2978 and 3649 are only 
probable primes.
%H A098088 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/66661.htm">Factorizations of 
66...661</a>.
%e A098088 If n = 4 we get ((2*10^4)-17)/3 = 19983/3 = 6661, which is prime.
%K A098088 more,nonn,new
%O A098088 0,1
%A A098088 Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

Form 6Rn+1
%I A096507
%S A096507 1,2,6,8,9,11,20,23,41,63,66,119,122,149,252,284,305,592,746,875,1204,
%T A096507 1364,2240,2403,5106,5776,5813,12456,14235
%N A096507 Exponents n for which 1+6*(10^n-1)/9 is a prime.
%C A096507 Same at n such that (2*10^n+1)/3 is prime.
%C A096507 All numbers from n = 2403 onwards are only probable primes. No others 
less than 25557. These numbers form a near-repdigit sequence (6)w7.
%e A096507 n = 9 gives 2000000001/3 = 666666667, which is prime.
%e A096507 n=20 means that 66666666666666666667 is a prime number.
%H A096507 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/66667.htm">Factorizations of 
66...667</a>.
%Y A096507 Cf. A096503-A096508.
%K A096507 base,nonn
%O A096507 0,2
%A A096507 Labos E. (labos(AT)ana1.sote.hu), Jul 12 2004
%E A096507 More terms from Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

This is the same A098087 so simply delete A098087.

Form 7Rn-6
%I A099419
%S A099419 2, 13, 20, 23, 31, 100, 241, 275, 925
%N A099419 Numbers n such that 7(10^n-1)/9 - 6 is prime.
%t A099419 Do[ If[ PrimeQ[ 7(10^n - 1)/9 - 6], Print[n]], {n, 0, 1000}]
%Y A099419 Cf. .
%O A099419 1,1
%K A099419 nonn
%A A099419 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 7Rn-4
%I A099420
%S A099420 1, 2, 3, 5, 9, 12, 15, 21, 264, 383
%N A099420 Numbers n such that 7(10^n-1)/9 - 4 is prime.
%t A099420 Do[ If[ PrimeQ[ 7(10^n - 1)/9 - 4], Print[n]], {n, 0, 1000}]
%Y A099420 Cf. .
%O A099420 1,2
%K A099420 nonn
%A A099420 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 7Rn+2
%I A098089
%S A098089 2,66,86,90,102,386,624
%N A098089 Integers n such that ((7*10^n)+11)/9 is prime.
%C A098089 n = 386 and n = 624 are only probably prime. The next term is greater 
than 5,000.
%H A098089 M. Kamada, <a 
href="http://homepage2.nifty.com/m_kamada/math/77779.htm">Factorizations of 
77...779</a>.
%e A098089 If n = 2, we get ((7*10^2)+11/9 = (700+11)/9 = 79, which is prime.
%t A098089 Do[ If[ PrimeQ[ 7(10^n - 1)/9 + 2], Print[n]], {n, 1000}] (from RGWv 
Oct 14 2004)
%K A098089 more,nonn,new
%O A098089 0,1
%A A098089 Julien Peter Benney (jpbenney(AT)ftml.net), Sep 14 2004

Form 8Rn-7
%I A099421
%S A099421 0, 3, 19, 79, 139, 223, 463, 544
%N A099421 Numbers n such that 8(10^n-1)/9 - 7 is prime.
%t A099421 Do[ If[ PrimeQ[ 8(10^n - 1)/9 - 7], Print[n]], {n, 0, 1000}]
%O A099421 0,2
%K A099421 nonn
%A A099421 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 8Rn-5
%I A099422
%S A099422 0, 1, 2, 3, 5, 8, 9, 15, 51, 71, 77, 224, 296, 315
%N A099422 Numbers n such that 8(10^n-1)/9 - 5 is prime.
%t A099422 Do[ If[ PrimeQ[ 8(10^n - 1)/9 - 5], Print[n]], {n, 0, 1000}]
%O A099422 0,3
%K A099422 nonn
%A A099422 Robert G. Wilson v (rgwv at rgwv.com), Oct 14 2004

Form 8Rn-1
%I A096846
%S A096846 1,3,4,6,9,12,72,118,124,190,244,304,357,1422
%N A096846 Exponents n for which -1+8*(10^n-1)/9 is a prime.
%e A096846 n=72: 
888888888888888888888888888888888888888888888888888888888888888888888887
%e A096846 is prime.
%Y A096846 Cf. A096503, A096504, A096505, A096506, A096507, A096508, A096841, 
A096842, A096843, A096844, A096845, A096846, A000203.
%K A096846 more,nonn
%O A096846 1,2
%A A096846 Labos E. (labos(AT)ana1.sote.hu), Jul 15 2004

Form 8Rn+1
%I A096508
%S A096508 2,14,17,35
%N A096508 Exponents n for which 1+8*(10^n-1)/9 is a prime.
%e A096508 n=35 means that 88888888888888888888888888888888889 is a prime number.
%t A096508 Do[ If[ PrimeQ[ 8(10^n - 1)/9 + 1], Print[n]], {n, 1000}] (from RGWv 
Oct 14 2004)
%Y A096508 Cf. A096503, A096504, A096505, A096506, A096507.
%K A096508 base,nonn
%O A096508 1,1
%A A096508 Labos E. (labos(AT)ana1.sote.hu), Jul 12 2004

Form 9Rn-8
%I A095714
%S A095714 3,5,7,33,45,105,197,199,281,301,317
%N A095714 k such that 10^k - 9 is a prime.
%e A095714 a(2) = 5, since 10^5 - 9 = 99991, which is prime
%t A095714 Select[ Range[ 1000], PrimeQ[10^# - 9] &]
%Y A095714 Cf. A088275.
%K A095714 nonn
%O A095714 0,1
%A A095714 Alonso Delarte (alonso.delarte(AT)gmail.com), Jul 07 2004

Form 9Rn-2
%I A089675
%S A089675 1,2,3,17,140,990,1887,3530,5996,13820,21873
%N A089675 Numbers n such that 10^n - 3 is a prime number.
%C A089675 Next term is greater than 22500. - Gabriel Cunningham 
(gcasey(AT)mit.edu), Mar 13 2004
%e A089675 10^2 - 3 = 97 is a prime number (in fact the largest less than 10^2).
%t A089675 To check for all n up to m: 
For[n=1,n<m,If[PrimeQ[10^n-3]==True,Print[n]];n++ ]
%K A089675 more,nonn
%O A089675 1,2
%A A089675 Michael Gottlieb (mzrg(AT)verizon.net), Jan 05 2004
%E A089675 a(8) from Robert G. Wilson v, Jan 14 2004.
%E A089675 a(9) and a(10) from Gabriel Cunningham (gcasey(AT)mit.edu), Mar 06 2004
%E A089675 a(11) from Gabriel Cunningham (gcasey(AT)mit.edu), Mar 13 2004

The 29 forms, X*10^n*Y*Repunits, (X,Y)=1,
			 X&Y belong to {d} d being the digits 1..9.

Form 1*10^n+1Rn
%I A004023 M2114
%S A004023 2,19,23,317,1031,49081,86453
%N A004023 Prime "repunits": 11...111 = (10^n - 1)/9 is prime.
............
%Y A004023 See A004022 for the actual primes.
%K A004023 hard,nonn,nice
%O A004023 1,1
%A A004023 njas

Form 2*10^n+1Rn
%I A056700
%S A056700 2,3,12,18,23,57,128,543,584,833
%N A056700 Numbers n such that 2*10^n + R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056700 Do[ If[ PrimeQ[ 2*10^n + (10^n-1)/9], Print[n]], {n, 0, 1538}]
%K A056700 hard,nonn
%O A056700 0,1
%A A056700 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 3*10^n+1Rn
%I A056704
%S A056704 1,2,5,10,11,13,34,47,52,77,88,554,580,1310,1505
%N A056704 Numbers n such that 3*10^n + 1*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056704 Do[ If[ PrimeQ[ 3*10^n + (10^n-1)/9], Print[n]], {n, 0, 1585}] n=9439 
also is prime.
%K A056704 hard,nonn
%O A056704 0,2
%A A056704 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 4*10^n+1Rn
%I A056706
%S A056706 1,3,13,25,72,108,375,393,589,973
%N A056706 Numbers n such that 4*10^n + 1*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056706 Do[ If[ PrimeQ[ 4*10^n + (10^n-1)/9], Print[n]], {n, 0, 1500}]
%K A056706 hard,nonn
%O A056706 0,2
%A A056706 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 5*10^n+1Rn
%I A056713
%S A056713 0,5,12,15,84,144,150,1235
%N A056713 Numbers n such that 5*10^n + R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056713 Do[ If[ PrimeQ[ 5*10^n + (10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056713 hard,nonn
%O A056713 0,2
%A A056713 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 6*10^n+1Rn
%I A056717
%S A056717 1,5,7,25,31,112,199,533,616,718,787,1357
%N A056717 Numbers n such that 6*10^n + R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056717 Do[ If[ PrimeQ[ 6*10^n + (10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056717 hard,nonn
%O A056717 0,2
%A A056717 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 7*10^n+1Rn
%I A056719
%S A056719 0,1,7,55
%N A056719 Numbers n such that 7*10^n + R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056719 Do[ If[ PrimeQ[ 7*10^n + (10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056719 hard,nonn
%O A056719 0,3
%A A056719 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 8*10^n+1Rn
%I A056722
%S A056722 2,3,26,110,141,474,902
%N A056722 Numbers n such that 8*10^n + R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056722 Do[ If[ PrimeQ[ 8*10^n + (10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056722 hard,nonn
%O A056722 0,1
%A A056722 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 9*10^n+1Rn
%I A056726
%S A056726 2,5,20,41,47,92,161,401,455
%N A056726 Numbers n such that 9*10^n + R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056726 Do[ If[ PrimeQ[ 9*10^n + (10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056726 hard,nonn
%O A056726 0,1
%A A056726 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 1*10^n+3Rn
%I A056698
%S A056698 1,15,41,83,95,341,551,669,989,1223
%N A056698 Numbers n such that 10^n + 3*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056698 Do[ If[ PrimeQ[ 10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 1570}]
%K A056698 hard,nonn
%O A056698 0,2
%A A056698 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 2*10^n+3Rn
%I A056701
%S A056701 0,1,2,3,4,10,16,22,53,91,94,106,138,210,282,522,597,1049
%N A056701 Numbers n such that 2*10^n + 3*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056701 Do[ If[ PrimeQ[ 2*10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 1500}]
%K A056701 hard,nonn
%O A056701 0,3
%A A056701 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 4*10^n+3Rn
%I A056707
%S A056707 1,2,16,31,37,55,62,172,174,197,727,1246
%N A056707 Numbers n such that 4*10^n + 3*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056707 Do[ If[ PrimeQ[ 4*10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 1500}]
%K A056707 hard,nonn
%O A056707 0,2
%A A056707 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 5*10^n+3Rn
%I A056714
%S A056714 0,1,3,13,25,49,143,419,1705
%N A056714 Numbers n such that 5*10^n + 3*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%C A056714 5*10^a(n)+3*(10^a(n)-1)/9 is a solution for part (b) of questions of 
puzzle 244 from www.primepuzzles.net. If a(n) is greater than 5812 then a(n) is an 
example that is asked for in this question. - Farideh Firoozbakht 
(f.firoozbakht(AT)sci.ui.ac.ir), Dec 02 2003
%H A056714 Prime Puzzles, <a 
href="http://www.primepuzzles.net/problems/prob_244.htm">Puzzle 244</a>
%t A056714 Do[ If[ PrimeQ[ 5*10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056714 hard,nonn
%O A056714 0,3
%A A056714 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000
%E A056714 1705 from Farideh Firoozbakht (f.firoozbakht(AT)sci.ui.ac.ir), Dec 18 2003

Form 7*10^n+3Rn
%I A056720
%S A056720 0,1,2,3,5,53,56,343,908,1079
%N A056720 Numbers n such that 7*10^n + 3*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056720 Do[ If[ PrimeQ[ 7*10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056720 hard,nonn
%O A056720 0,3
%A A056720 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 8*10^n+3Rn
%I A056723
%S A056723 1,7,23,29,133,173,367
%N A056723 Numbers n such that 8*10^n + 3*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056723 Do[ If[ PrimeQ[ 8*10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056723 hard,nonn
%O A056723 0,2
%A A056723 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 1*10^n+7Rn
%I A089147
%S A089147 1,3,9,13,42,51,54,91,120,168,510,819,1071,1756
%N A089147 Numbers n such that 1 followed by n 7's is prime.
%t A089147 Do[ If[ PrimeQ[10^n + 7(10^n - 1)/9], Print[ n]], {n, 2100}]
%Y A089147 For the primes see A088465.
%K A089147 base,nonn
%O A089147 1,2
%A A089147 Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 05 2003

Form 2*10^n+7Rn
%I A056702
%S A056702 0,2,3,9,15,18,36,63,114,225,405,482,1241
%N A056702 Numbers n such that 2*10^n + 7*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056702 Do[ If[ PrimeQ[ 2*10^n + 7*(10^n-1)/9], Print[n]], {n, 0, 1500}]
%K A056702 hard,nonn
%O A056702 0,2
%A A056702 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 3*10^n+7Rn
%I A056705
%S A056705 0,1,11,17,773
%N A056705 Numbers n such that 3*10^n + 7*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056705 Do[ If[ PrimeQ[ 3*10^n + 7*(10^n-1)/9], Print[n]], {n, 0, 1500}]
%K A056705 hard,nonn
%O A056705 0,3
%A A056705 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 4*10^n+7Rn
%I A056708
%S A056708 1,4,13,25,36,357,373,1041,1089,1093,1297
%N A056708 Numbers n such that 4*10^n + 7*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056708 Do[ If[ PrimeQ[ 4*10^n + 7*(10^n-1)/9], Print[n]], {n, 0, 1500}]
%K A056708 hard,nonn
%O A056708 0,2
%A A056708 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 5*10^n+7Rn
%I A056715
%S A056715 0,2,8,14,17,18,33,35,126,183,324,344,866,992,1226
%N A056715 Numbers n such that 5*10^n + 7*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056715 Do[ If[ PrimeQ[ 5*10^n + 7*(10^n-1)/9], Print[n]], {n, 0, 2000}]
%K A056715 hard,nonn
%O A056715 0,2
%A A056715 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 6*10^n+7Rn
%I A056718
%S A056718 1,2,4,10,13,25,115,179,181,238,785,799,1193
%N A056718 6*10^n + 7*R_n, where R_n = 11...1 is the repunit (A002275) of length n.
%t A056718 Do[ If[ PrimeQ[ 6*10^n + 7*(10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056718 hard,nonn
%O A056718 0,2
%A A056718 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 8*10^n+7Rn
%I A056724
%S A056724 2,9,15,32,38,65,123,173,257,320,326,639,719,774,902
%N A056724 Numbers n such that 8*10^n + 7*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056724 Do[ If[ PrimeQ[ 8*10^n + 7*(10^n-1)/9], Print[n]], {n, 0, 1600}]
%K A056724 hard,nonn
%O A056724 0,1
%A A056724 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 9*10^n+7Rn
%I A056727
%S A056727 1,2,4,19,28,73,203,220,274,292,470,763,1891
%N A056727 Numbers n such that 9*10^n + 7*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056727 Do[ If[ PrimeQ[ 9*10^n + 7*(10^n-1)/9], Print[n]], {n, 0, 1600}]
%Y A056727 Cf. A093944 (corresponding primes).
%K A056727 hard,nonn
%O A056727 0,2
%A A056727 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 1*10^n+9Rn
%I A002957 M0680
%S A002957 1,2,3,5,7,26,27,53,147,236,248,386,401,546,785,1325,1755,2906,3020
%N A002957 Numbers n such that 2*10^n - 1 is prime.
%C A002957 n such that 10^n + 9*R_n is prime, where R_n = 11...1 is the repunit 
(A002275) of length n.
%D A002957 H. Riesel, "Prime numbers and computer methods for factorization," 
Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Page 162.
%D A002957 C. R. Zarnke and H. C. Williams, Computer determination of some large 
primes, pp. 563-570 in Proceedings of the Louisiana Conference on Combinatorics, 
Graph Theory and Computer Science. Vol. 2, edited R. C. Mullin et al., 1971.
%t A002957 Do[ If[ PrimeQ[ 2*10^n - 1], Print[n] ], {n, 1, 4800} ]
%K A002957 hard,nonn
%O A002957 1,2
%A A002957 njas, Simon Plouffe (plouffe(AT)math.uqam.ca)
%E A002957 Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 02 
2001. n=7517 also produces a prime.

Form 2*10^n+9Rn
%I A056703
%S A056703 0,1,3,6,7,19,27,43,55,207,1311
%N A056703 Numbers n such that 2*10^n + 9*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056703 Do[ If[ PrimeQ[ 2*10^n + (10^n-1)], Print[n]], {n, 0, 1500}] n=9439 
also is prime.
%K A056703 hard,nonn
%O A056703 0,2
%A A056703 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 10 2000

Form 4*10^n+9Rn
%I A056712
%S A056712 2,3,4,6,14,54,210,390,594
%N A056712 Numbers n such that 4*10^n + 9*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056712 Do[ If[ PrimeQ[ 4*10^n + (10^n-1)], Print[n]], {n, 0, 1600}] Note: for 
n = 8072 also produces a prime.
%K A056712 hard,nonn
%O A056712 0,1
%A A056712 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 5*10^n+9Rn
%I A056716
%S A056716 0,1,2,4,5,7,10,13,22,23,28,34,40,61,73,361,490,613,1624,2000,2994,4301,
%T A056716 4332
%N A056716 Numbers n such that 6*10^n-1 is prime.
%C A056716 Next term is > 15000. - Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 12 2004
%t A056716 Do[ If[ PrimeQ[ 6*10^n - 1], Print[n]], {n, 0, 5000}]
%Y A056716 Cf. A056805 (6*10^n+1 is prime).
%K A056716 hard,nonn
%O A056716 0,3
%A A056716 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000
%E A056716 More terms from Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 12 2004

Form 7*10^n+9Rn
%I A056721
%S A056721 0,1,4,5,8,10,25,49,76,128,175,238,550,796,1219
%N A056721 Numbers n such that 7*10^n + 9*R_n is prime, where R_n = 11...1 is the 
repunit (A002275) of length n.
%t A056721 Do[ If[ PrimeQ[ 7*10^n + (10^n-1)], Print[n]], {n, 0, 1600}]
%K A056721 hard,nonn
%O A056721 0,3
%A A056721 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000

Form 8*10^n+9Rn
%I A056725
%S A056725 1,3,7,19,29,37,93,935
%N A056725 Numbers n such that 9*10^n - 1 is prime.
%C A056725 8*10^n + 9*R_n is prime, where R_n = 11...1 is the repunit (A002275) of 
length n.
%t A056725 Do[ If[ PrimeQ[ 8*10^n + (10^n-1)], Print[n]], {n, 1, 5900, 2}]
%Y A056725 Cf. A003307, A002235, A046865, A079906, A046866, A001771, A005541, 
A046867, A079907.
%K A056725 hard,nonn
%O A056725 0,2
%A A056725 Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 11 2000
%E A056725 There are no more terms up to 6750. - Robert G. Wilson v, Jan 26 2003

Repunit (1)w
Near-repdigit (R)wD
Near-repdigit D(R)w
Near-repdigit Palindrome (R)wD(R)w
Plateau and Depression D(R)wD
Generalized quasi-repdigit D(R)wE